2.2 Core Band Energy Dispersion
27
2.2 Core Band Energy Dispersion
The following single-body Hamiltonian and eigen wavefunction describe an electron
moving in the νth orbit of an atom in the ideal bulk [9],
H = H 0 + H
with
H 0 = −
2 ∇
2
2m
+ V atom (r ) (Total energy for an electron of an isolated atom)
H = V cr yst (r )
( Inter - atomic interaction)
|ν, i ∼ = u(r ) exp(ikr)
( Bloch wave - function)
(2.1)
H 0 is the Hamiltonian for an isolated atom, which sums the kinetic energy and
intra-atomic potential energy experienced by the specific electron. The interatomic
potential V cryst (r) sums all interactions with neighboring atoms and electrons. There
are might be multiple constituents for the V cryst (r) but the spectroscopy collect them
inclusively in a convoluted form without needing any decomposition. The V atom (r)
= V atom (r + R) < 0, the V cryst (r) = V cryst (r + R) < 0, and the Bloch wavefunctions
are periodic in real space, where R is the lattice constant. Because of the localization nature of the core electrons, the eigen wavefunction |ν, i meets the following
criterion, where i and j denote atomic positions:
ν, j |ν, i = δ i j =
1 (i = j)
0 (i = j)
The coupling of the potentials experienced by an electron represented by the and
its Bloch wavefunction determines the energy shift. The energy of an electron in
an ideal bulk disperses in the following manner (with atomic CN or z b = 12 for an
fcc-structured bulk standard):
E ν (z b ) = E ν (0) + (α ν + z b β ν ) + 2z b β ν ν (k, R)
with
⎧
⎨
⎩
E ν (0) = −ν, i|H 0 |ν, i
(Atomic core level)
α ν
= −
ν, i
H
ν, i
∝ E b (Exchange integral)
β ν
= −
ν, i
H
ν, j
∝ E b (Overlap integral)
(2.2)
The V atom (r) defines the energy level of an isolated atom E ν (0), from which the
core band shifts. As intrinsic constants, the E ν (0) reduces its value with the quantum
number ν from 10
3 to 10
0 eV until the vacuum level E 0 = 0 as the ν increases, or as
one moves from the innerest orbit outwardly of an atom.
The CLS fingerprints the variation of interatomic interaction that changes with
chemical and coordination environment. The involvement of the V cryst (r) upon bulk or
liquid formation deepens the E ν (0) by an amount of E ν (z b ) = E ν (z b ) − E ν (0) =
α ν + z b β ν , and meanwhile, turns the CL into a band of E νW = 2z b β ν ν (k, R)
width. Both the exchange integral α ν and the overlap integral β ν are proportional to
the cohesive energy per bond at equilibrium, E b , or the zeroth approximation of the
27
2.2 Core Band Energy Dispersion
The following single-body Hamiltonian and eigen wavefunction describe an electron
moving in the νth orbit of an atom in the ideal bulk [9],
H = H 0 + H
with
H 0 = −
2 ∇
2
2m
+ V atom (r ) (Total energy for an electron of an isolated atom)
H = V cr yst (r )
( Inter - atomic interaction)
|ν, i ∼ = u(r ) exp(ikr)
( Bloch wave - function)
(2.1)
H 0 is the Hamiltonian for an isolated atom, which sums the kinetic energy and
intra-atomic potential energy experienced by the specific electron. The interatomic
potential V cryst (r) sums all interactions with neighboring atoms and electrons. There
are might be multiple constituents for the V cryst (r) but the spectroscopy collect them
inclusively in a convoluted form without needing any decomposition. The V atom (r)
= V atom (r + R) < 0, the V cryst (r) = V cryst (r + R) < 0, and the Bloch wavefunctions
are periodic in real space, where R is the lattice constant. Because of the localization nature of the core electrons, the eigen wavefunction |ν, i meets the following
criterion, where i and j denote atomic positions:
ν, j |ν, i = δ i j =
1 (i = j)
0 (i = j)
The coupling of the potentials experienced by an electron represented by the and
its Bloch wavefunction determines the energy shift. The energy of an electron in
an ideal bulk disperses in the following manner (with atomic CN or z b = 12 for an
fcc-structured bulk standard):
E ν (z b ) = E ν (0) + (α ν + z b β ν ) + 2z b β ν ν (k, R)
with
⎧
⎨
⎩
E ν (0) = −ν, i|H 0 |ν, i
(Atomic core level)
α ν
= −
ν, i
H
ν, i
∝ E b (Exchange integral)
β ν
= −
ν, i
H
ν, j
∝ E b (Overlap integral)
(2.2)
The V atom (r) defines the energy level of an isolated atom E ν (0), from which the
core band shifts. As intrinsic constants, the E ν (0) reduces its value with the quantum
number ν from 10
3 to 10
0 eV until the vacuum level E 0 = 0 as the ν increases, or as
one moves from the innerest orbit outwardly of an atom.
The CLS fingerprints the variation of interatomic interaction that changes with
chemical and coordination environment. The involvement of the V cryst (r) upon bulk or
liquid formation deepens the E ν (0) by an amount of E ν (z b ) = E ν (z b ) − E ν (0) =
α ν + z b β ν , and meanwhile, turns the CL into a band of E νW = 2z b β ν ν (k, R)
width. Both the exchange integral α ν and the overlap integral β ν are proportional to
the cohesive energy per bond at equilibrium, E b , or the zeroth approximation of the